Equivalent resistance between the adjacent corners of a regular $$n$$-sided polygon of uniform wire of resistance $$R$$ would be :

The wire has a total resistance $$R$$ and is divided into $$n$$ equal sides. Therefore, the resistance of each side ($$r$$) is:

$$r = \frac{R}{n}$$

Path 1 (Minor Arc), consists of only 1 side. $$R_1 = r = \frac{R}{n}$$

Path 2 (Major Arc), consists of the remaining $$(n-1)$$ sides connected in series. $$R_2 = (n-1)r = \frac{(n-1)R}{n}$$

Using the parallel formula $$R_{eq} = \frac{R_1 \times R_2}{R_1 + R_2}$$:

$$R_{eq} = \frac{\left( \frac{R}{n} \right) \left( \frac{(n-1)R}{n} \right)}{\frac{R}{n} + \frac{(n-1)R}{n}}$$

$$R_{eq} = \frac{(n-1)R}{n^2}$$